Theorema egregium

The Theorema egregium is a sentence from differential geometry , a branch of mathematics . It was found by Carl Friedrich Gauß and in a nutshell it reads:

The Gaussian curvature of a surface is a quantity of the internal geometry of . ${\displaystyle S\subset \mathbb {R} ^{3}}$ ${\displaystyle S}$

The Gaussian curvature is one of the most important curvature quantities in classical differential geometry. The theorema egregium follows from Brioschi's formula .

history

While Gauß had measured the Kingdom of Hanover from 1821 to 1825, he suspected that the curvature of the earth's surface could be determined solely by measuring length and angle. In fact, it took Gauss some time to prove this statement. His proof was anything but straightforward and simple. For this reason he called the sentence the egregium theorema , "outstandingly important theorem".

Classification in modern differential geometry

The differential geometry experienced by Gauss essential impulses. This led to the fact that later the curvature considered by Gauss was also called Gaussian curvature. In addition, one can consider that the length and angle measurement on a surface is induced by the coefficients of the first fundamental form . In the language of differential geometry, the statement of the Theorema egregium reads:

The Gaussian curvature depends only on the coefficients of the matrix of the first fundamental form and their first and second derivatives.

In this sense, the Gaussian curvature is a quantity of the inner geometry, i.e. the geometry that is only induced by the first fundamental form. Further variables of the internal geometry are the length measurement of a curve of the surface, the area and also the geodetic curvature of a curve.

Derivation

As already mentioned, Gauss himself was only able to determine this rate after a lengthy calculation. Later these calculations could be simplified considerably. For example, Brioschi's formula applies:

${\displaystyle K={\frac {1}{(EG-F^{2})^{2}}}\left({\begin{vmatrix}-{\frac {1}{2}}E_{vv}+F_{uv}-{\frac {1}{2}}G_{uu}&{\frac {1}{2}}E_{u}&F_{u}-{\frac {1}{2}}E_{v}\\F_{v}-{\frac {1}{2}}G_{u}&E&F\\{\frac {1}{2}}G_{v}&F&G\end{vmatrix}}-{\begin{vmatrix}0&{\frac {1}{2}}E_{v}&{\frac {1}{2}}G_{u}\\{\frac {1}{2}}E_{v}&E&F\\{\frac {1}{2}}G_{u}&F&G\end{vmatrix}}\right)}$

Where , and are the coefficients of the first fundamental form with respect to a parameterization . The terms , etc. stand for the first and second partial derivatives according to the parameters and , with which the given area is parameterized. This formula is proven by applying the definition of Gaussian curvature, the multiplication formula for determinants and a nifty representation of the higher derivatives of the position vector of the surface by coefficients of the first fundamental form. ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle G}$ ${\displaystyle (u,v)\mapsto \varphi (u,v)}$ ${\displaystyle E_{u}}$ ${\displaystyle F_{uv}}$ ${\displaystyle u}$ ${\displaystyle v}$

The theorema egregium follows from this as a corollary .

literature

• Carl Friedrich Gauß : Disquisitiones generales circa superficies curvas (General investigations on curved surfaces; October 8, 1827), Commentationes Societatis Regiae Scientiarum Gottingensis recentiores 6 (classis mathematicae), 1828, pp. 99–146 (beginning p. 313 in the scanned document) , and Dieterich, Gottingae (Göttingen) 1828 (Latin; Theorema egregium on p. 120 or p. 24: [1] ; also in Carl Friedrich Gauß: Werke. Volume 4 , pp. 219–258, Theorema egregium on p. 237 )
• Wilhelm Blaschke , Kurt Leichtweiß: Elementary differential geometry . Basics of mathematical Sciences. Volume 1. Springer, Berlin 1973, ISBN 0-387-05889-3 .
• Wilhelm Klingenberg : A lecture on differential geometry . Springer, Berlin 1973, ISBN 3-540-06253-X .
• Manfred P. do Carmo: Differential geometry of curves and surfaces . Vieweg, Braunschweig 1993, ISBN 3-528-27255-4 .
• Peter Dombrowski : Differential Geometry - 150 years after the "Disquisitiones generales circa superficies curvas" by Carl Friedrich Gauß . Treatises of the Braunschweigische Wissenschaftliche Gesellschaft Volume 27, 1977, pp. 63-102.

Web links

• Mathematics - a human mind's eye On the historical background from a lecture by Eberhard Zeidler at the DMV conference in 2006
• The page on MathWorld